Internal problem ID [3554]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 11
Problem number: 298.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime }-n \left (y^{2}-2 y x +1\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 279
dsolve((-x^2+1)*diff(y(x),x) = n*(1-2*x*y(x)+y(x)^2),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {x +1}{x -1}\right )^{n} \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 n} \left (16 \left (x +1\right )^{2} \left (\left (x -\frac {1}{2}\right ) n +\frac {1}{2}-\frac {x}{2}\right ) \operatorname {hypergeom}\left (\left [-n +1, -n +1\right ], \left [2-2 n \right ], -\frac {2}{x -1}\right ) c_{1} \left (\frac {x +1}{x -1}\right )^{-n}+\left (x -1\right ) \left (\left (x +1\right )^{2} n \left (\frac {x +1}{x -1}\right )^{n} \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n} \operatorname {hypergeom}\left (\left [n , n\right ], \left [2 n \right ], -\frac {2}{x -1}\right )-16 \left (\frac {\operatorname {HeunCPrime}\left (0, 2 n -1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) \left (x +1\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n}}{8}+\operatorname {HeunCPrime}\left (0, -2 n +1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1} \right ) \left (x -1\right )\right )\right )}{\left (x +1\right )^{2} \left (8 c_{1} \operatorname {hypergeom}\left (\left [-n +1, -n +1\right ], \left [2-2 n \right ], -\frac {2}{x -1}\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 n}+\left (\frac {x +1}{x -1}\right )^{2 n} \operatorname {hypergeom}\left (\left [n , n\right ], \left [2 n \right ], -\frac {2}{x -1}\right ) \left (x -1\right )\right ) n} \]
✓ Solution by Mathematica
Time used: 0.361 (sec). Leaf size: 47
DSolve[(1-x^2)*y'[x]==n*(1-2*x*y[x]+y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {Q_n(x)+c_1 P_n(x)}{Q_{n-1}(x)+c_1 P_{n-1}(x)} \\ y(x)\to \frac {P_n(x)}{P_{n-1}(x)} \\ \end{align*}